Problem: Find the slope and y-intercept of the line that is ${\text{perpendicular}}$ to $\enspace {y = \dfrac{1}{2}x + 4}\enspace$ and passes through the point ${(-6, 7)}$. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$
Lines are considered perpendicular if their slopes are negative reciprocals of each other. The slope of the blue line is ${\dfrac{1}{2}}$ , and its negative reciprocal is ${-2}$ Thus, the equation of our perpendicular line will be of the form $\enspace {y = -2x + b}\enspace$ We can plug our point, $(-6, 7)$ , into this equation to solve for ${b}$ , the y-intercept. $7 = {-2}(-6) + {b}$ $7 = 12 + {b}$ $7 - 12 = {b} = -5$ The equation of the perpendicular line is $\enspace {y = -2x - 5}\enspace$. ${m = -2, \enspace b = -5}$